1. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

2. If \( \cos\theta = p \) and \( \cot\theta = q \), then which of the following relationships is true?

(a) \(\cfrac{1}{p^2}+\cfrac{1}{q^2}=1\) (b) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=1\) (c) \(\cfrac{1}{p^2}-\cfrac{1}{q^2}=0\) (d) \(\cfrac{1}{q^2}+\cfrac{1}{p^2}=1\)

3. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct?

(a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)

4. If \(x = a\sec\theta\) and \(y = b\tan\theta\), then find the relation between \(x\) and \(y\) eliminating \(\theta\).

5. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

6. If \( \tan^2\theta + \cot^2\theta = \cfrac{10}{3} \), then find the values of \( \tan\theta + \cot\theta \) and \( \tan\theta - \cot\theta \). From there, determine the value of \( \tan\theta \).

7. If \(3x = 5\sin\theta\) and \(4y = 5\cos\theta\), then which of the following relations is correct?

(a) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=1\) (b) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=0\) (c) \(\cfrac{9x^2}{25}-\cfrac{16y^2}{25}=1\) (d) \(\cfrac{16x^2}{25}+\cfrac{9y^2}{25}=1\)

8. If \(∑f_i(x_i - a) = 400\), \(∑f_i = 50\), and \(a\) (assumed mean) = 52, then the value of the combined mean \(\bar{x}\) is –

(a) 52 (b) 60 (c) 80 (d) 90

9. If \(∑f_i d_i = 400\), \(∑f_i = 50\), and \(a =\) assumed mean \(= 52\), then the value of the combined mean is –

(a) 52 (b) 60 (c) 80 (d) 55

10. If \(tanθ = 0\), then which of the following is correct?

(a) sinθ=0 (b) cosθ=0 (c) cotθ=0 (d) None of these

11. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

12. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

13. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

14. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

15. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

16. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

17. \(\theta\) is a positive acute angle, and if \( \tan\theta = \cot\theta \), then what is the value of \(\theta\)?

(a) 40° (b) 45° (c) 60° (d) 20°

18. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

19. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?

20. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?

(a) 0 (b) 1 (c) 0.1 (d) 0.01

21. If \(u_i = \cfrac{x_i - 20}{10}\), \(\sum{f_iu_i} = 15\), and \(\sum{f_i} = 80\), then what will be the value of \(\bar{x}\)?

(a) 21.875 (b) 20.875 (c) 21.800 (d) 20.125

22. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?

(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)

23. If \(\sum\limits_{i=1}^5 x_i = 5\) and \(\sum\limits_{i=1}^5 x_i^2 = 14\), then the value of \(\sum\limits_{i=1}^5 2x_i(x_i - 3)\) will be —

(a) 2 (b) -2 (c) 0 (d) 4

24. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

25. In right-angled triangle ABC, \(\angle B = 90^\circ\), and M is the midpoint of the hypotenuse drawn from point B. If AB = 6 and AC = 8, then what is the length of BM?

(a) 10 (b) 4 (c) 16 (d) None of the above

26. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?

27. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

28. If \(\cos \theta = \frac{x}{y}\) and \(x \ne y\), then which one is smaller between \(x\) and \(y\), and why?

29. If \(m = \sqrt{\cfrac{n}{n + \cfrac{1}{2}}}\) and \(m = \cfrac{1}{2}\), then what is the value of \(n\)?

30. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then find the simplest value of \(\frac{x^2 - xy + y^2}{x^2 + xy + y^2}\).